Question

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = x³ − 6x² + 11x − 6, g(x) = x² + x + 1

Answer 1

We have

f(x) = x³ − 6x² + 11x − 6

g(x) = x² + x + 1

Here, degree [f(x)] = 3 and

Degree (g(x)) = 2

Therefore, quotient q(x) is of degree 3 - 2 = 1 and the remainder r(x) is of degree less than 2

Let q(x) = ax + b and

r(x) = cx + d

Using division algorithm, we have

f(x) = g(x) x q(x) + r(x)

x³ − 6x² + 11x − 6 = (x² + x + 1)(ax + b) + cx + d

x³ − 6x² + 11x − 6 = ax³ + ax² + ax + bx² + bx + b + cx + d

x³ − 6x² + 11x − 6 = ax³ + ax² + bx² + ax + bx + cx + b + d

x³ − 6x² + 11x − 6 = ax³ + (a + b)x² + (a + b = c)x + b + d

Equating the co-efficients of various powers of x on both sides, we get

On equating the co-efficient of x³

x³ = ax³

1 = a

On equating the co-efficient of x²

-6x² = (a + b)x²

-6 = a + b

Substituting a = 1

-6 = 1 + b

-6 - 1 = b

-7 = b

On equating the co-efficient of x

11x = (a + b + c)x

11 = a + b + c

Substituting a = 1 and b = -7 we get,

11 = 1 + (-7) + c

11 = -6 + c

11 + 6 = c

17 = c

On equating the constant terms

-6 = b + d

Substituting b = -7 we get,

-6 = -7 + d

-6 + 7 = d

1 = d

Therefore, 

Quotient q(x) = ax + b

= (1x - 7)

And remainder r(x) = cx + d

= (17x + 1)

Hence, the quotient and remainder is given by,

q(x) = (x - 7)

r(x) = 17x + 1